Long division of polynomials is an important concept for students to learn not only because it is a standard in the curriculum but because it serves as a foundation in various other fields of mathematics and because it has significant applications in many careers that students may pursue in the future.
For instance, dividing polynomials may be used in data transmission such as error correction and encryption as well as in the fields of chemical engineering and pharmaceutical development.
The following article describes how dividing polynomials helps build towards more advanced topics in calculus and other upper level math courses, while also detailing how it may be used in the real world.
There are at least two separate areas where the insights gained from understanding the long
division algorithm are crucial, first calculus, and second, applications of polynomials in
advanced areas of mathematics and related fields.
Let us discuss calculus first. The student's experience of carrying long division past the
decimal point is his or her first experience with infinite processes of any kind converging.
At each stage the accuracy of the answer increases approximately by a factor of 10.
Compare this with the only other infinite process we can reasonably assume students have
experienced, that of counting numbers. At each stage we're no closer to the end than the
stage before. Polynomial long division leads in a natural way both to the geometric series
and to the priliminary expansion (1 – xn+1) = (1 – x)( 1 + x + x2 + x3 + … + xn ).
The development of polynomial long division up to this point is modeled on the assumption
that higher powers of x are larger than lower powers, just as 1015 is larger than 105 .
However, the reverse assumption is often equally valid – that higher powers of x become
much smaller than lower powers, as, for example happens when x = 0.1. This leads to a
second approach to long division for polynomials. For example, consider the problem of
dividing 1 by (1 – x). By iterating the long division process, using higher powers of x as
remainders, we obtain the expression
1/(1-x) = 1 + x + x2 + x3 + … + xn with remainder xn+1.
This gives a basic formula with innumerable applications. For example, it provides the
underpinnings of compound interest, and further in the future, functions such as the
exponential function. (The exponential function arises by letting the time between
compounding periods approach zero, or, as one might say, considering continuous
compounding.)
More advanced topics also depend on basic polynomial manipulations of the type described
above. An early application in the university curriculum occurs in calculus courses, where
students learn how to use the partial fraction decomposition to integrate rational functions
(quotients of polynomials). Previous to this the students have learned the rules for
integrating simple polynomials and some trigonometric functions, all of which are quite
direct, following basically from the definitions and the fundamental theorem of calculus.
But it turns out – unfortunately only much later – that the partial fraction decompositions
play a unique and important role in modern applications of mathematics. They form the
basis for the techniques needed for using the Laplace Transform in handling systems of
linear differential equations, and consequently for many areas of engineering. Indeed,
engineers in any area which involves control systems – aeronautical engineering, mechanical
engineering, and much of electrical engineering – will confirm that most of what they do
depends on Laplace Transforms.
There is reason to believe that students who wish to enter technical areas but have had only
minimal experience with polynomials and none with partial fraction techniques are at a
severe disadvantage, and at least some are forced to give up their aspirations.
Of course, the applications to the Laplace Transform discussed above are not the only, nor
the best known uses of basic polynomial operations in vital applications of mathematics.
The other major area of application is linear algebra. Here, manipulations involving the
characteristic polynomial are critical in determining the eigenvalues and eigenvectors of
linear transformations, which in turn have basic applications to economics, the social
sciences, as well as the physical sciences and engineering.
In order to engage the students and increase their motivation to learn about dividing polynomials, teachers may present problems and utilize strategies such as these that illustrate the real world applications of this topic.
Offer the students a simple example with a discussion about two-dimensional construction planning. For example, you could ask the students to create a garden and give them the total amount of available soil and the width of the garden. Then ask them to calculate the garden's length using the polynomial function.
For students planning to enter the legal or financial planning industry, many of the interest rate formulas are polynomial equations. For example, have a student calculate the value of his college savings account with a deposit amount of $200. Using the standard annual interest calculation of 2,000(1+r)^3, have the student calculate what value the account will have in three years.
Introduce the students to the complexities of architecture and how polynomials are applied in that field by showing them a video about how New York's Trump Tower was created and the application of polynomials with that building in a video clip from Thefutureschannel.com, an educational tools website.
Entice your students with basic aerospace science through Newtonian physics problems such as the equations for objects in motion, projectiles and even air resistance.
Long division of polynomials is an important concept for students to learn not only because it is a standard in the curriculum but because it serves as a foundation in various other fields of mathematics and because it has significant applications in many careers that students may pursue in the future.
For instance, dividing polynomials may be used in data transmission such as error correction and encryption as well as in the fields of chemical engineering and pharmaceutical development.
The following article describes how dividing polynomials helps build towards more advanced topics in calculus and other upper level math courses, while also detailing how it may be used in the real world.
The Role of Long Division in the K-12 Curriculum
There are at least two separate areas where the insights gained from understanding the long
division algorithm are crucial, first calculus, and second, applications of polynomials in
advanced areas of mathematics and related fields.
Let us discuss calculus first. The student's experience of carrying long division past the
decimal point is his or her first experience with infinite processes of any kind converging.
At each stage the accuracy of the answer increases approximately by a factor of 10.
Compare this with the only other infinite process we can reasonably assume students have
experienced, that of counting numbers. At each stage we're no closer to the end than the
stage before. Polynomial long division leads in a natural way both to the geometric series
and to the priliminary expansion (1 – xn+1) = (1 – x)( 1 + x + x2 + x3 + … + xn ).
The development of polynomial long division up to this point is modeled on the assumption
that higher powers of x are larger than lower powers, just as 1015 is larger than 105 .
However, the reverse assumption is often equally valid – that higher powers of x become
much smaller than lower powers, as, for example happens when x = 0.1. This leads to a
second approach to long division for polynomials. For example, consider the problem of
dividing 1 by (1 – x). By iterating the long division process, using higher powers of x as
remainders, we obtain the expression
1/(1-x) = 1 + x + x2 + x3 + … + xn with remainder xn+1.
This gives a basic formula with innumerable applications. For example, it provides the
underpinnings of compound interest, and further in the future, functions such as the
exponential function. (The exponential function arises by letting the time between
compounding periods approach zero, or, as one might say, considering continuous
compounding.)
More advanced topics also depend on basic polynomial manipulations of the type described
above. An early application in the university curriculum occurs in calculus courses, where
students learn how to use the partial fraction decomposition to integrate rational functions
(quotients of polynomials). Previous to this the students have learned the rules for
integrating simple polynomials and some trigonometric functions, all of which are quite
direct, following basically from the definitions and the fundamental theorem of calculus.
But it turns out – unfortunately only much later – that the partial fraction decompositions
play a unique and important role in modern applications of mathematics. They form the
basis for the techniques needed for using the Laplace Transform in handling systems of
linear differential equations, and consequently for many areas of engineering. Indeed,
engineers in any area which involves control systems – aeronautical engineering, mechanical
engineering, and much of electrical engineering – will confirm that most of what they do
depends on Laplace Transforms.
There is reason to believe that students who wish to enter technical areas but have had only
minimal experience with polynomials and none with partial fraction techniques are at a
severe disadvantage, and at least some are forced to give up their aspirations.
Of course, the applications to the Laplace Transform discussed above are not the only, nor
the best known uses of basic polynomial operations in vital applications of mathematics.
The other major area of application is linear algebra. Here, manipulations involving the
characteristic polynomial are critical in determining the eigenvalues and eigenvectors of
linear transformations, which in turn have basic applications to economics, the social
sciences, as well as the physical sciences and engineering.
In order to engage the students and increase their motivation to learn about dividing polynomials, teachers may present problems and utilize strategies such as these that illustrate the real world applications of this topic.
Read more: How to Use Polynomials in the Real World | eHow.com http://www.ehow.com/how_5968709_use-polynomials-real-world.html#ixzz1lILS3jFF